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Mathematics

Quarter 1 – Module 1:

Generating Patterns and

Illustrating Arithmetic Sequence

Mathematics – Grade 10

Alternative Delivery Mode

Quarter 1 – Module 1 : Generating Patterns and Illustrating Arithmetic Sequence

First Edition, 2020

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Office Address: Provincial Capitol Compound, Balanga City, Bataan

Telefax: (04 7 ) 237 - 2102

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Development Team of the Module

Writer: Flordeliza R. Angeles

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Reviewer: Sherwin G. Serrano

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Rufino V. Rubino

ii

Introductory Message

For the facilitator:

Welcome to the Mathematics – Grade 10 Alternative Delivery Mode (ADM)

Module on Generating Patterns and Illustrating Arithmetic Sequence!

This module was collaboratively designed, developed and reviewed by

educators both from public and private institutions to assist you, the teacher or

facilitator in helping the learners meet the standards set by the K to 12 Curriculum

while overcoming their personal, social, and economic constraints in schooling.

This learning resource hopes to engage the learners into guided and

independent learning activities at their own pace and time. Furthermore, this also

aims to help learners acquire the needed 21st century skills while taking into

consideration their needs and circumstances.

In addition to the material in the main text, you will also see this box in the

body of the module:

As a facilitator you are expected to orient the learners on how to use this

module. You also need to keep track of the learners' progress while allowing them

to manage their own learning. Furthermore, you are expected to encourage and

assist the learners as they do the tasks included in the module.

Notes to the Teacher

This contains helpful tips or strategies that

will help you in guiding the learners.

iii

For the learner:

Welcome to the Mathematics – Grade 10 Alternative Delivery Mode (ADM)

Module on Generating Patterns and Illustrating Arithmetic Sequence!

The hand is one of the most symbolized part of the human body. It is often

used to depict skill, action and purpose. Through our hands we may learn, create

and accomplish. Hence, the hand in this learning resource signifies that you as a

learner is capable and empowered to successfully achieve the relevant

competencies and skills at your own pace and time. Your academic success lies in

your own hands!

This module was designed to provide you with fun and meaningful

opportunities for guided and independent learning at your own pace and time. You

will be enabled to process the contents of the learning resource while being an

active learner.

This module has the following parts and corresponding icons:

What I Need to Know

This will give you an idea of the skills or

competencies you are expected to learn in the

module.

What I Know

This part includes an activity that aims to

check what you already know about the

lesson to take. If you get all the answers

correct (100%), you may decide to skip this

module.

What’s In

This is a brief drill or review to help you link

the current lesson with the previous one.

What’s New

In this portion, the new lesson will be

introduced to you in various ways such as a

story, a song, a poem, a problem opener, an

activity or a situation.

What is It

This section provides a brief discussion of the

lesson. This aims to help you discover and

understand new concepts and skills.

What’s More

This comprises activities for independent

practice to solidify your understanding and

skills of the topic. You may check the

answers to the exercises using the Answer

Key at the end of the module.

What I Have Learned

This includes questions or blank

sentence/paragraph to be filled in to process

what you learned from the lesson.

What I Need to Know

One of the most amazing things we can observe from our environment is that it is

full of patterns and sequences. The way petals of flowers are arranged, designs in

our floor tiles, the cone of pine tree and even the outside appearance of pineapple

fruit, these all exhibit patterns.

Around the world, police departments have relied on mathematics in solving some of

their cases. Special algorithm can use the information about the past crimes to

predict on when or where crimes might occur. It also seemed like earthquakes follow

the same pattern just as crimes. An earthquake might trigger an aftershock just like

a crime might result to another crime of retaliation.

The above scenario presented an ideal chance for the learners to realize that studying

patterns are important. This scenario illustrates a sequence. In this learning module,

you will know more about sequences and how the concept of a sequence is utilized

in our daily lives.

In these lessons you will learn to:

1. generate and describe patterns. (M10AL-Ia- 1 )
2. illustrates an arithmetic sequence. (M10AL-Ib- 1 )

What I Know

Directions. Find out how much you already know about the lessons in this module.

Choose the letter of the best answer.

1. Complete the following pattern by filling in the blanks and then describe the

pattern in words.

B B G B G B Y B B G __ G B __ B B __ B G B __ __

a. B, Y, G, Y, B b. G, B, B, Y, G c. Y, Y, G, G, B d. B, G, G, Y,

2. Look at the pattern below. Continue the pattern by filling in the blanks.

O, T, T, F, F, S, S, E, N, T, E, T, T, __, __, __, __

a. F, F, T, F b. S, O, F, T c. F, F, S, S d. S, S, F, F

Explain how the change is created in the following patterns and sequence.

a. Add 6 c. subtract 6

b. Divide by 2 d. Multiply by 2

a. Subtract 5 c. Add 5

b. Subtract 3 d. Add 3

a. Add 4 c. Subtract 4

b. Multiply by 4 d. Divide by 4

6. Determine which is NOT an arithmetic sequence in the choices below.

a. 1,2,3,4,5 c. 3,9,27,

b. 4.0, 4.5, 5.0, 5.5 d. 13, 2, - 9, - 20, - 31

7. What is the next term of the following arithmetic sequence, - 5, - 1, 3,7,11…?

a. 12 b. 13 c. 14 d. 15

8. Which of the statement below is not true about this arithmetic sequence?

a. The 7

th

term is 60 c. The common difference is 7

b. The 6

th

term is 60 d. The general rule is a n

= 18 +7n

9. Give the first three terms of the nth term, a n

= 2n-1.

a. 2, 3, 4 b. 3, 4 ,5 c. 1, 3, 5 d. 3, 5, 7

10. Formulate the rule for the given sequence, 10, 17, 24, 31, …

a. a n

= 8n- 4 c. a n

= 5n+

b. a n

= 7n+3 d. a n

= 4n+

11. What comes next? A, 2, B, 4, C, 6, __, ___, …

a. D, 10 b. B, 8 c. B, 10 d. D, 10

12. The first term of an arithmetic sequence is 3 while the 7

th

term is 21, what is

the common difference?

a. 3 b. 4 c. 5 d. 6

13. Solve for the 20

th

term of the arithmetic sequence, - 12, - 4, 4, 12 …

a. 120 b. 130 c. 140 d. 150

14. Calculate the nth term of the arithmetic sequence, 2, 6, 4, 10, …

a. 3n+2 b. 4n- 2 c. n+2 d. n- 2

15. Solve for the first three terms of the arithmetic sequence using the nth term,

a n

3 𝑛+ 5

2

a. 4.5, 6, 8.5 b. 4, 5.5, 7 c. 4, 5,5, 7.5 d. 5, 7.5, 9

Instructions. Check your answers after you have finished answering the items

above. (Refer to the answer key at the back matters for correct answers.) If you get

100% correct, you can skip the module. If 50% to 99% correct, you have to proceed

with the module.

What’s New

Below is an activity. In this activity you will work with pattern recognition.

Activity 1. Each item below shows a pattern. Take this test as you would take a test

in class. Then check your work with the solutions in the answer key at the

back matters.

1. What is the next shape?

, , , , , , , , , , , , , , __.

2. 0 , 3 , 6 , 9 , 12 , ___.

What is the next number?

What is the 10

th

number?

3. 7 , 3 , - 1 , - 5 , - 9 , __.

What is the next number?

What is the 9

th

number?

4. 1 , 4 , 16 , 64 , ___.

What is the next number?

What is the 10

th

number?

5. 120 , 60 , 30 , 15 , ___.

What is the next number?

What is the 7

th

number?

In the next items, draw the fourth object following the pattern.

, , , ____________

, , , _____________

, , , _____________

9. , , , ____________

, , , _______________

How did you find the activity? Were you able to find the patterns and get the next

number in the sequence?

What is It

Let’s now give the formal definition of a sequence. The set of figures and numbers

above the given activities are called sequences. A special notation is often used with

sequence. Instead of writing 𝑎( 3 ) = 6 to indicate the 3

rd

term, we write 𝒂

𝟑

= 𝟔. This

is read as “ 𝑎 𝑠𝑢𝑏 3 𝑒𝑞𝑢𝑎𝑙𝑠 6 .” The number 3 is the index because it indicates the

position of the term in a sequence.

A sequence (of real numbers) is a function whose domain is the finite set {1, 2, 3,

... 𝑛 } or infinite set { 1 , 2 , 3 ,... }.

Set of ordered pair numbers can also be written in tabular form.

Finite set

This is a finite sequence that has 5 terms {0, 3, 6, 9, 12}. The pattern used to get the

succeeding term is 𝒂 𝒏

= 𝟑𝒏 − 𝟑. (Steps in forming this pattern will be discuss to you

later.)

Infinite set

This is an infinite sequence that has an infinite number of terms denoted by three

dots (…), the pattern used to get the succeeding term is 𝑎 𝑛

In the next activity, you will learn more about sequences. A general term or n

th

term will be given to you as a guide to solve the next few terms.

Before you proceed, here are the examples.

Example 1

Find the first 5 terms of the sequence with the given n

th

rule as your guide.

The first 5 terms (1, 2, 3, 4, 5) are to be substituted one at a time into the n

th

term.

The n

th

term is 𝒂

𝒏

𝑛

1

2

2

2

2

1

2

2

2

2

Therefore, the first 5 terms of the sequence using the n

th

rule 𝒂

𝒏

= 𝒏 + 𝟒 are 5,

Here is another example:

Example 2

Find the first 5 terms of the sequence with the given n

th

term 𝑎

𝑛

Again, substitute (1, 2, 3, 4, 5) one at a time into the n

th

rule 𝑎

𝑛

n 1 2 3 4 5

𝑛

n 1 2 3 4 …

𝑛

n 1 2 3 4 5

𝑛

How did you find the activity? Did you find it easy to give the first 5 terms of each

sequences? In the next activity, you will be given the terms of a sequence and you

will find its n

th

term.

The next activity that you will do, is the reverse of Activity 2. You will be given sets

of sequences then you will give the general term or the n

th

term.

Example 1

Find the n

th

term of the sequence 5, 7, 9, 11…

Solution: Draw a table of values.

where 𝑛 are the first 𝑛 terms

𝑛

is the n

th

term of the sequence

First, get the common difference. To find the common difference, subtract the

succeeding term from the preceeding term.

Example: 7 9 11

- 5 _ - _ 7 _ - _ 9 _

In this example, the common difference is 2, multiply this by 𝑛 and add/subtract a

particular number to get 𝑎

𝑛

. In this example, the particular number is 3. Therefore,

the generated pattern or n

th

term 𝒂

𝒏

= 𝟐𝒏 + 𝟑. To check, substitute in place of 𝑛, the

first 𝑛 terms (1, 2, 3, 4) using the generated pattern or n

th

term 𝒂

𝒏

Solution:

𝒏

Given: 𝑛 = { 1 , 2 , 3 , 4 } 𝑎 1

2

3

4

1

2

3

4

1

2

3

4

2. What is the 15

th

term of the sequence 5, 7, 9, 11?

Given: 𝑛 = 15

Using the n

th

term: 𝒂

𝒏

= 𝟐𝒏 + 𝟑 therefore, the 15

th

term of the

15

= 2 ( 15 ) + 3 sequence 5, 7, 9, 11 is 33.

15

15

Formulating a generated pattern or the n

th

term is useful because it lets you

calculate a specific term without having to calculate all the previous terms.

Example 2

What is the n

th

term of the sequence - 2, 5, 12, 19?

Solution:

1. Draw the table of values.

2. Find the common difference 𝒅 = 𝟕.

3. The particular number to add/subtract is - 9.

n 1 2 3 4

𝑛

n 1 2 3 4

𝑛

Therefore: 𝑎

𝑛

To check, substitute the first n terms (1, 2, 3, 4) one at a time in place of 𝑛.

a) 𝑎

1

= 7 ( 1 ) − 9 b) 𝑎

2

= 7 ( 2 ) − 9 c) 𝑎

3

= 7 ( 3 ) − 9 d) 𝑎

4

1

2

3

4

1

2

3

4

Example 3

What is the 10

th

term of the sequence?

𝑛

10

10

10

After giving you the three (3) examples, you can now proceed to the next activity.

Find the generated pattern or the n

th

term of the given sequences below.

The first five terms of the sequence are 1, 2, 3, 4, 5.

Note: Take this activity as you would take an activity in class. Check your work

with solutions found in the answer key in the back matter.

In the activities you have just done, you were able to enumerate the terms of a

sequence, given its n

th

term and vice versa. Knowing these will enable you to easily

understand a particular sequence, the arithmetic sequence.

What I Have Learned

This lesson is about sequences which involve generating and describing patterns

using symbols and mathematical expressions. These patterns are used in finding the

next few terms, and the n

th

term of the given sequence. The lesson provides the

students with opportunities to illustrate sequences using practical situation. The

students are given the chance to create sequences as illustrated in some real-life

situations. Their understanding of this lesson will help them to understand and learn

the next lesson, Arithmetic Sequence.

Complete the following sentences to make the statement true.

1. A sequence is a function whose domain is the __________________ set or

______________________ set.

2. Set of ordered pair of number can also be written in __________________________

form.

3. To find the __________________________________, subtract the succeeding term

from the preceding term.

Assessment

I. Choose the letter that you think best answers the question.

1. What is missing in the sequence ___, 4, 10, 16, ___?

a. 2 and 22 b. - 2 and 22 c. 2 and 24 d. - 2 and 24

2. Give the first three terms of the 𝑛

th

term 𝑎

𝑛

a. 10, 12, 15 b. 10, 13, 16 c. 10, 12, 14 d. 10, 11, 12

3. Formulate the rule for the given sequence, 9, 6, 3, 0, - 3.

a. 𝑎

𝑛

= 12 − 3 𝑛 b. 𝑎

𝑛

= 3 𝑛 + 12 c. 𝑎

𝑛

= 3 𝑛 + 6 d. 𝑎

𝑛

4. What is the next term in the sequence, 4, 7, 10, 13, 16, 19, __?

a. 20 b. 21 c. 22 d. 23

5. Find the next term, 7, 15, 23, 31 and the common difference.

a. 38; 5 b. 38; 6 c. 39; 7 d. 39; 8

6. What is the n

th

term of the sequence, 2, 4, 6, 8, …?

a. 𝑎

𝑛

= 𝑛 + 1 b. 𝑎

𝑛

2

• 1 c. 𝑎

𝑛

= 2 𝑛 d. 𝑎

𝑛

7. What are the next two terms in the sequence, 3, 6, 9, 12, ___?

a. 14; 16 b. 15; 16 c. 14; 15 d. 15; 18

8. What is the 7

th

term of the sequence, 4, 10, 16, …?

a. 20 b. 30 c. 40 d. 50

9. Formulate the n

th

term of the sequence, 7, 15, 23, 31 …

a. 𝑎

𝑛

= 8 𝑛 − 1 b. 𝑎

𝑛

= 8 𝑛 + 1 c. 𝑎

𝑛

= 7 𝑛 − 1 d. 𝑎

𝑛

10. The first 3 terms of the sequence given the rule 𝑎

𝑛

𝑛

a. 1, 2, 4 b. 2, 4, 8 c. 2, 6, 8 d. 2, 4, 6

II. On a number square like this one, shade all the multiples of 7. Then

answer the questions that follows.

11. What is the 3

rd

multiple of 7?

12. What is the 10

th

multiple of 7?

13. What is the 24

th

multiple of 7?

14. What is the 50

th

multiple of 7?

15. What is the 100

th

multiple of 7?

Additional Activities

A. The Missing Link!

Try to fill up the missing numbers below and determine the numerical sequence;

1. 15, 20, 25, 30, 35, 40, ____, ____, 55, ____, 65, 70, 75, ____, 85, ____

The rule for this numerical sequence is: __________

2. 93, 87, 81, ____, ____, 63, 57, 51, 45, ____, 33, 27, 21, ____, ____, 3

The rule for this numerical sequence is: __________

3. Generate two numerical sequences starting at zero using the given rules.

Then compare and explain the relationship between the two sequences.

Add 2: , , , , ,

Add 8: , , , , ,

4. Generate two numerical sequences starting at zero using the given rules.

Then compare and explain the relationship between the two sequences.

Add 3: , , , , ,

Add 27: , , , , ,

B. Decide whether a sequence is arithmetic or not. If it is, find the common

difference.

Example: 3, 7, 11, 15, 19,...

a.) arithmetic sequence 𝑑 = 4

1

2

1

2

Compare and explain:

Compare and explain:

References

Melvin M. Callanta, et. Al, 2015, Mathematics Learner’s Module 10, 1

st

ed.

Pasig City, Philippines 1600: Rex Bookstore, Inc., pp. 6- 15

Melvin M. Callanta, et. Al, 2015, Mathematics Teacher’s Guide 10, 1

st

ed.

Pasig City, Philippines 1600: Rex Bookstore, Inc., pp. 13 - 17

https://mathigon.org/course/sequences/introduction

https://www.helpingwithmath.com/printables/worksheets/algebra/5oa3patterns

3.htm

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Department of Education – Region III,

Schools Division of Bataan - Curriculum Implementation Division

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Provincial Capitol Compound, Balanga City, Bataan

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Email Address: [email protected]